Reconstruction of the Space-dependent Source from Partial Neumann Data for Slow Diffusion System

Consider a linear inverse problem of determining the space-dependent source term in a diffusion equation with time fractional order derivative from the flux measurement specified in partial boundary. Based on the analysis on the forward problem and the adjoint problem with inhomogeneous boundary condition, a variational identity connecting the inversion input data with the unknown source function is established. The uniqueness and the conditional stability for the inverse problem are proven by weak unique continuation and the variational identity in some norm. The inversion scheme minimizing the regularizing cost functional is implemented by using conjugate gradient method, with numerical examples showing the validity of the proposed reconstruction scheme.